Issue 59

A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03

T

 m Q PQ P

(9)

where P is the transformation matrix given by:                                    2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 P

(10)

     cos ;

     sin and

    2 sin 2 and 

m Q is the stress-reduced stiffness for the material plane

where

      1 12 2 12 2 2

0 0 0 0 0 0

       

       

G



0 0 0

0 0 0

0 0

Q

(11)

23

m

G

0

0

13

G

0 0

12

E

E

1

2

Here 2 E is the equivalent Young’s modulus for the layers that are perpendicular to fibres. On the other hand,  12 and  21 are Poisson’s ratios. 23 G is the shear modulus for planes 2 − 3, 13 G is the equivalent shear modulus for planes 1 − 3, and 12 G represent the shear modulus for planes 1 − 2. The variables w ,  x and  y can be expressed in the harmonic forms as follows:                            , , , , , , , , , i t i t x x i t w x y t x y e x y t x y e x y t x y e (12)        1 12 21 1 ; and    2     12 21 1 , where 1 E is Young’s modulus for a layer parallel to fibres. And



y

y

and the equations of motion (6) become:

          

   

   

  

  

  

2

2

 



 







  2

x

x







 

K A

A

I

 

s

55

44

0

2

2





x

x



x

y

   

   

    2

2

2

2

   

 





 

  

y

y

2

x

 D D







      I

D

K A

(13)

 

s

x

x

11

12

66

55

2

2

2

 



x y

x y

x



x

y

   

   

  2

  2

  2

    2

  

  



 

y

y

    2 I

x

x











D

D

D

K A

y

s

y

2

12

22

66

44

2

2

2



x y

y







y

y

x

In which  is natural frequency.

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